Low degree rational spline interpolation

For a strictly monotone functionf on [a,b] we describe the possibility of finding an interpolating rational splineS of the formS(x)=c ₀ +c ₁ x/(1+d ₁ x) on each subinterval of the grida=x ₀ <x ₁ <...<x _n =b. This leads to a nonlinear system for which we get the local existence and uniqueness of a solution. We prove that ‖S−f‖_∞=O(h ³). Numerical test shows good approximation properties of these splines.     

Monotonicity preserving rational spline histopolation

For given monotone data we propose the construction of a histopolating linear/linear rational spline of class C¹. The uniqueness and existence of this spline is proved. The method is implemented via the representation with histogram heights and first derivatives of the spline. The use of Newton's method and ordinary iterations are discussed. Numerical tests support completely the theoretical results.     

The spline curve,a smooth interpolating function used in numerical design of ship-lines

It has been found that a numerical approximation of a ship-designer's spline results in a smooth interpolating function. An algorithm which computes the function coefficients is presented in Algol. Methods to vary the stiffnes of the spline are included. A test of fairness is described and finally a more elaborate algorithm which leads to a closer approximation is outlined.     

Error estimation of a kind of rational spline

This paper deals with the approximation properties of a kind of rational spline with linear denominator when the function being interpolated is C³$C^3$ in an interpolating interval. Error estimate expressions of interpolating functions are derived, convergence is established, the optimal error coefficient, _ci$c_i$, is proved to be symmetric about the parameters of the rational interpolation and it is bounded. Finally, the precise jump measurements of the second derivatives of the interpolating function at the knots are given.     

Shape preserving C2 cubic spline interpolation

A globally C² interpolatory cubic spline containing free parametersis derived and its properties established. Sufficient conditionsare given for choosing the parameters to control monotonicity;convexity is discussed. An algorithm is developed and testedon several examples.     

Optimal designs for weighted rational models

This paper is concerned with constructing optimal designs for rational models which are used for modeling problems in Agriculture and other disciplines. Homoscedastic and weighted models are considered. An analytical characterization of these designs is obtained as zeros of a polynomial solution of a second order differential equation.     

Hermite G 1 rational spline motion of degree six

Applying geometric interpolation techniques to motion construction has many advantages, e.g., the parameterization is chosen automatically and the obtained rational motion is of the lowest possible degree. In this paper a G ¹ Hermite rational spline motion of degree six is presented. An explicit solution of nonlinear equations that determine the spherical part of the motion is derived. Particular emphasis is placed on the construction of the translational part of the motion. Since the center trajectory is a G ¹ continuous for an arbitrary choice of lengths of tangent vectors, additional free parameters are obtained, which are used to minimize particular energy functionals. Thenumerical examples provide an evidence that the obtained motions have nice shapes.     

On weighted approximation by rational operators for functions with singularities

We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [?1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia?[5].     

Decentralized control by rational controllers

A strongly decentralized control problem is considered where only local information is available on both states and goals (the desired states) of each subsystem. A theory is presented which allows one to optimize the behavior of a system with such an information structure.The development is based on a notion of rational controllers which are ergodic dynamical systems with some additional property of optimality. Analysis of the interacting rational controllers gives the conditions under which a system of rational controllers also exhibits a rational behavior. The knowledge of these conditions allows one to suggest a control technique for a strongly decentralized situation.Dedicated to R. Bellman     

Cam design by hyperbolic spline functions of fourth order

E-mail: kohaupt{at}tfh-berlin.de In this paper, cam design using splines is discussed. The splinesare of 4th order so as to guarantee the continuity of the valve-liftcurve up to the third derivative in order to obtain valves withbetter dynamic behaviour. First, ordinary polynomial splinesare investigated, and their limited potential in cam designis illustrated. Then, the reason for introducing the new hyperbolicsplines is explained, and their potential for cam design isshown. Finally, specific cam designs based on the new splinesare compared with cams developed by Kurz. It turns out thatspline cams are an interesting alternative to Kurz's designsince—in some cases—they show a better dynamic behaviourwhen compared with Kurz's cams. The spline cam has been employed,e.g. in the M103 engine of Mercedes-Benz.     

Cubic algebraic spline curves design

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G ² continuous contact with the control polygon at two endpoints and is G ² continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.     

Approximation by polynomials and rational functions in weighted rearrangement invariant spaces

Let Γ be a Dini-smooth curve in the complex plane, and let G:=IntΓ. We prove some direct and inverse theorems of approximation theory by algebraic polynomials and rational functions in the weighted rearrangement invariant Smirnov spaces EX(G,ω) defined on G.     

A multiresolution analysis for tensor-product splines using weighted spline wavelets

We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing-as a model problem-image compression with black-and-white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.     

Automorphisms of real rational surfaces and weighted blow-up singularities

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e = [e ₁, . . . , e _? ] be a partition of n. Denote by X ^e the set of ?-tuples (P ₁, . . . , P _? ) of disjoint nonsingular curvilinear subschemes of X of orders (e ₁, . . . , e _? ). We show that the group Aut(X) acts transitively on X ^e . This statement generalizes earlier work where the case of the trivial partition e = [1, . . . , 1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.     

Equivariant cohomology of certain moduli of weighted pointed rational curves

We determine the action of the product of symmetric groups on the cohomology of certain moduli of weighted pointed rational curves. The moduli spaces that we study are of stable rational curves with m + n marked points where the first m marked points are distinct from all the others where as the last n may coincide among themselves. We give a recipe for calculating the equivariant Poincaré polynomials and list them for small m and n.